Pub Date : 2019-07-18DOI: 10.1061/9780784409213.apb
René Orth
Linear Algebra Methods in Combinatorics This file contains solutions to some of the exercises, and it will be periodically updated. 1 Exercise Set 2 Exercise 4 (One-Distance Sets). A regular simplex is a set of n + 1 points in R n such that any two points are at distance 1. Prove that no set with this property can have more points. n are such that d(s i , s j) = 1 for all i = j (1) where d() is the Euclidean distance, then m ≤ n + 1. If you cannot prove this bound, try to prove the simpler bound m ≤ n + 2. Warm up (" n + 2 "). We can assume wlog that d() is the square of the Euclidean distance (this will not change the problem), that is d(x, y) = k
本文件包含部分习题的解答，并会定期更新。1练习2练习4(单距离组)正则单纯形是rn中n + 1个点的集合，使得任意两个点的距离为1。证明具有这个性质的集合不可能有更多的点。对于所有i = j(1)，其中d()为欧氏距离，则m≤n + 1，则d(s i, s j) = 1。如果你不能证明这个界，试着证明更简单的界m≤n + 2。热身(“n + 2”)。我们可以假设log d()是欧氏距离的平方(这不会改变问题)也就是d(x, y) = k
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Pub Date : 2019-07-18DOI: 10.1017/9781108668804.008
ly speaking, the universally-quantified predicate corresponds to the subsheaf given by the following pullback:
一般来说，普遍量化的谓词对应于以下回拉给出的子轴:
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Pub Date : 2019-07-01DOI: 10.1017/9781108668804.004
Brendan Fong, David I. Spivak
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Pub Date : 2019-07-01DOI: 10.1017/9781108668804.005
Brendan Fong, David I. Spivak
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